Perturbation Analysis for the Drazin Inverse Under Stable Perturbation in Banach Space

Abstract

Let $X$ be Banach space and let $T, \bar T=T+\delta T$ be bounded linear operators on $X$. Suppose that $T$ has the Drazin inverse $T^D$ and $\text{Ind} (T)=n$. In this paper, we show that if $\|\delta T\|$ is sufficiently small and $\text{Ran} (\bar T^n) \cap \text{Ker} (({T^D})^n) = \{0\}$, then $\bar T$ is Drazin invertible with $\text{Ind} (\bar T)\le n$. In this case, the expression of $\bar T^D$ is given and the upper bounds of $\|\bar T^D\|$ and $$\dfrac{\|\bar T^D-T^D\|}{\|T^D\|}$$ are established. If $\dim X<\infty$, replacing $\text{Ran} (\bar T^n) \cap \text{Ker} (({T^D})^n) = \{0\}$ by $\text{rank} (\bar T^n) = \text{rank} (T^n)$, we obtain the same perturbation results of the Drazin invertible matrix $T$ as in the case of $\dim X=\infty$.